A homography is a linear projective transformation relating two images for either of two cases: (1) a planar world object projected onto image planes for the two images, where the image planes relate to arbitrary camera motion; or (2) the projection of arbitrary world images onto image planes for pure camera rotation only. A pinhole camera model illustrated in FIG. 4A is employed for the first view to define the location within the camera reference frame (x, y, and z, where z is the optical axis) of an image point m representing the intersection of a line containing a three dimensional world point W and the camera's optical center C with an image plane R. The projection of the optical center C′ for a similar pinhole camera representing the second view onto the image plane R of the first camera or view, illustrated in FIG. 4B, is epipole e, and epipole e′ is similarly the projection of the optical center C onto the image plane R′ of the second camera or view.
The image points m (defined by coordinates x,y) and m′ (defined by coordinates x′,y′), which represent the projection of the image pixel W onto the image planes R and R′, respectively, constitute a conjugate pair, and a fundamental matrix F relating such conjugate pairs may be computed (see, e.g., “Epipolar Geometry and the Fundamental Matrix” in R. Hartley & A. Zisserman, Multiple View Geometry in Computer Vision, pp. 219–242 (Cambridge University Press 2000), incorporated herein by reference) and employed to develop a 3×3 homography matrix H12 for projective mapping of image points from the first camera or view reference frame to the second according to the linear projective equation:             [                                                  x              ′                                                                          y              ′                                                            1                              ]        =                  [                                                            h                11                                                                    h                12                                                                    h                13                                                                                        h                21                                                                    h                22                                                                    h                23                                                                                        h                31                                                                    h                32                                                                    h                33                                                    ]            ⁢                          [                                    x                                                y                                                1                              ]        ,which may be equivalently written as:                                           x            ′                    =                                                                      h                  11                                ⁢                x                            +                                                h                  12                                ⁢                y                            +                              h                13                                                                                      h                  31                                ⁢                x                            +                                                h                  32                                ⁢                y                            +                              h                33                                                                                              y            ′                    =                                                                      h                  21                                ⁢                x                            +                                                h                  22                                ⁢                y                            +                              h                23                                                                                      h                  31                                ⁢                x                            +                                                h                  32                                ⁢                y                            +                              h                33                                                          .Homography describes a one-to-one point mapping between the two images for either pure rotation by the camera or for world image points within a plane. The homography matrix for the plane at infinity H12∞, which maps vanishing points to vanishing points, is an important case and depends only on the rotational component of the rigid displacement.
Computation of homography is a very important and difficult task in, for example, three dimensional (3D) video generation and rendering. This is especially true for computation of the homography matrix for the plane at infinity, which requires knowledge of either (1) three sets or parallel lines, (2) point matches between the views corresponding to points very far away from the camera, or (3) complete camera calibration parameters. None of these is easy to compute for the general case.
There is, therefore, a need in the art for a simplified method of computing the homography matrix for the plane at infinity, particularly in extended image sequences.